From Newsgroup: rec.puzzles


Pondering the Eight Location Puzzle reminded me of a nice Geometric Gem.

How many ways are there to arrange 4 points in the plane such that
only two inter-point distances occur. For example, one solution is the
four corners of a unit square: Among the six point-to-point edge lengths,
four are sides with length 1 and two are diagonals with length 1.41421356.

For our purpose here, two arrangements are IDENTICAL if one can be
converted into the other via translation, rotation, reflection or resizing.

Which arrangement is hardest to think of?

Cheers,
James
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From Newsgroup: rec.puzzles

On Tue, 17 Mar 2026 19:14:20 GMT, James Dow Allen <user4353@newsgrouper.org.invalid> wrote:

Pondering the Eight Location Puzzle reminded me of a nice Geometric Gem.

How many ways are there to arrange 4 points in the plane such that
only two inter-point distances occur. For example, one solution is the

I came up with two more ...... but cannot think of any others.

......

SPOILERS BELOW


.....

SPOILERS BELOW


SPOILERS BELOW


.....

SPOILERS BELOW

SPOILERS BELOW


.....

SPOILERS BELOW

SPOILERS BELOW


.....

SPOILERS BELOW


The two have to do with rather rudimentary logic.
What if one starts with three points equidistant
from each other -- an equilateral triangle.

One solution instantly falls out. The other one
is a variant of the same. It plays on the old
geometry things of points "within" and "without"
a closed figure.

Now, to rack my brains some more!
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From Newsgroup: rec.puzzles


Charlie Roberts <croberts@gmail.com> posted:

On Tue, 17 Mar 2026 19:14:20 GMT, James Dow Allen <user4353@newsgrouper.org.invalid> wrote:

Pondering the Eight Location Puzzle reminded me of a nice Geometric Gem.

How many ways are there to arrange 4 points in the plane such that
only two inter-point distances occur. For example, one solution is the

I should have specified that the points must be DISTINCT.
Otherwise there are four solutions where one of the distances is zero!

A closely related puzzle is to arrange FIVE distinct points such that
only two inter-point distances occur.

(SIX points with THREE distances? No. NO-NO-NO.
At some point fun becomes masochism.
Fbzr lbhat ynqvrf va Oreyva gnhtug zr gung.)

I came up with two more ...... but cannot think of any others.

The two have to do with rather rudimentary logic.
What if one starts with three points equidistant
from each other -- an equilateral triangle.

One solution instantly falls out. The other one
is a variant of the same. It plays on the old
geometry things of points "within" and "without"
a closed figure.

Now, to rack my brains some more!

Come on, folks: Help Charlie here!
(I'm beginning to think r.p is moribund.)
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From Newsgroup: rec.puzzles

On Wed, 18 Mar 2026 16:36:57 -0400, Charlie Roberts
<croberts@gmail.com> wrote:

Came up with one more!

On Tue, 17 Mar 2026 19:14:20 GMT, James Dow Allen ><user4353@newsgrouper.org.invalid> wrote:

Pondering the Eight Location Puzzle reminded me of a nice Geometric Gem.

How many ways are there to arrange 4 points in the plane such that
only two inter-point distances occur. For example, one solution is the

I came up with two more ...... but cannot think of any others.

......

SPOILERS BELOW

.....

SPOILERS BELOW

SPOILERS BELOW

.....

SPOILERS BELOW

SPOILERS BELOW

.....

SPOILERS BELOW

SPOILERS BELOW

.....

SPOILERS BELOW

The two have to do with rather rudimentary logic.
What if one starts with three points equidistant
from each other -- an equilateral triangle.

One solution instantly falls out. The other one
is a variant of the same. It plays on the old
geometry things of points "within" and "without"
a closed figure.

Now, to rack my brains some more!

This one popped up when I was NOT thinking
about the puzzle!!

It is a variant of the equilateral triangle solutions
above. The fourth point still lies on one of the
angular biscetors of the original equilateral
triangle. The distance from the vertex that
is bisected is the side of the original equilateral
triangle.

That makes four, in total, solutions.


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From Newsgroup: rec.puzzles

On 20/03/2026 15:04, Charlie Roberts wrote:

On Wed, 18 Mar 2026 16:36:57 -0400, Charlie Roberts
<croberts@gmail.com> wrote:

Came up with one more!

On Tue, 17 Mar 2026 19:14:20 GMT, James Dow Allen
<user4353@newsgrouper.org.invalid> wrote:

Pondering the Eight Location Puzzle reminded me of a nice Geometric Gem. >>>
How many ways are there to arrange 4 points in the plane such that
only two inter-point distances occur. For example, one solution is the

I came up with two more ...... but cannot think of any others.

......

SPOILERS BELOW


.....

SPOILERS BELOW


SPOILERS BELOW


.....

SPOILERS BELOW

SPOILERS BELOW


.....

SPOILERS BELOW

SPOILERS BELOW


.....

SPOILERS BELOW


The two have to do with rather rudimentary logic.
What if one starts with three points equidistant
from each other -- an equilateral triangle.

One solution instantly falls out. The other one
is a variant of the same. It plays on the old
geometry things of points "within" and "without"
a closed figure.

Now, to rack my brains some more!

This one popped up when I was NOT thinking
about the puzzle!!

It is a variant of the equilateral triangle solutions
above. The fourth point still lies on one of the
angular biscetors of the original equilateral
triangle. The distance from the vertex that
is bisected is the side of the original equilateral
triangle.

There are two such solutions, depending on the direction taken from the vertex. So, that makes 5
solutions?

Mike.

That makes four, in total, solutions.

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From Newsgroup: rec.puzzles

On Fri, 20 Mar 2026 06:03:14 GMT, James Dow Allen <user4353@newsgrouper.org.invalid> wrote:

Charlie Roberts <croberts@gmail.com> posted:

On Tue, 17 Mar 2026 19:14:20 GMT, James Dow Allen
<user4353@newsgrouper.org.invalid> wrote:

How many ways are there to arrange 4 points in the plane such that
only two inter-point distances occur. For example, one solution is the

I should have specified that the points must be DISTINCT.
Otherwise there are four solutions where one of the distances is zero!

I came up with two more ...... but cannot think of any others.


The two have to do with rather rudimentary logic.
What if one starts with three points equidistant
from each other -- an equilateral triangle.

One solution instantly falls out. The other one
is a variant of the same. It plays on the old
geometry things of points "within" and "without"
a closed figure.

Now, to rack my brains some more!

Come on, folks: Help Charlie here!
(I'm beginning to think r.p is moribund.)

Well, not thinking helps (sometimes!). One more
solutuion came to me. Again, it deals with an
equilaternal triangle and an angular bisector. But,
this time, bisect an external angle and on the
bisector, find a point such the the distance between
that point and the bisected vertex is equal to the
sides of the original equilateral triangle.

Voila!

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From Newsgroup: rec.puzzles

On 3/17/2026 12:14 PM, James Dow Allen wrote:

Pondering the Eight Location Puzzle reminded me of a nice Geometric Gem.

How many ways are there to arrange 4 points in the plane such that
only two inter-point distances occur. For example, one solution is the
four corners of a unit square: Among the six point-to-point edge lengths, four are sides with length 1 and two are diagonals with length 1.41421356.

For our purpose here, two arrangements are IDENTICAL if one can be
converted into the other via translation, rotation, reflection or resizing.

Which arrangement is hardest to think of?

Cheers,
James

Partial Spoiler...




Partial Spoiler...




Partial Spoiler...




Partial Spoiler...




Partial Spoiler...




Partial Spoiler...




Partial Spoiler...





For me, the hardest of the six was the second arrangement that does not include an equilateral triangle.
--
Carl G.


--
This email has been checked for viruses by AVG antivirus software.
www.avg.com
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From Newsgroup: rec.puzzles


"Carl G." <carlgnews@microprizes.com> posted:

On 3/17/2026 12:14 PM, James Dow Allen wrote:

How many ways are there to arrange 4 points in the plane such that
only two inter-point distances occur?

For me, the hardest of the six was the second arrangement that does not include an equilateral triangle.

And it is Carl G. who successfully completes the sextet.
This solution -- which Mr. G carefully leaves unspoiled -- is surely
the hardest to come up with.

Now: Solve the companion puzzle:
James Dow Allen <user4353@newsgrouper.org.invalid> posted:

A closely related puzzle is to arrange FIVE distinct points such that
only two inter-point distances occur.

... and describe the weird/wonderful relationship between this 5-point companion puzzle and the question I asked earlier:

Which of the six {4-point, 2-distances} arrangements is hardest to think of?

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From Newsgroup: rec.puzzles

On 20/03/2026 17:11, James Dow Allen wrote:

"Carl G." <carlgnews@microprizes.com> posted:

On 3/17/2026 12:14 PM, James Dow Allen wrote:

How many ways are there to arrange 4 points in the plane such that
only two inter-point distances occur?

For me, the hardest of the six was the second arrangement that does not
include an equilateral triangle.

And it is Carl G. who successfully completes the sextet.
This solution -- which Mr. G carefully leaves unspoiled -- is surely
the hardest to come up with.

Indeed. I see the solution now. It's the only one whose geometric compass-and-straightedge
construction isn't obvious. (Euclid knew how to do it, but I doubt your average school leaver would
get very far!)

Mike.

Now: Solve the companion puzzle:
James Dow Allen <user4353@newsgrouper.org.invalid> posted:

A closely related puzzle is to arrange FIVE distinct points such that
only two inter-point distances occur.

... and describe the weird/wonderful relationship between this 5-point companion puzzle and the question I asked earlier:

Which of the six {4-point, 2-distances} arrangements is hardest to think of?

I think this 5-point puzzle is likely to be easier for many people than finding the 6'th solution to
the 4-point puzzle, and might lead them to the 6'th solution indirectly! (That's amusing when you
think about it...)

Anyway, nice puzzle!

Mike.






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From Newsgroup: rec.puzzles


Mike Terry <news.dead.person.stones@darjeeling.plus.com> posted:

On 20/03/2026 17:11, James Dow Allen wrote:

Now: Solve the companion puzzle:
James Dow Allen <user4353@newsgrouper.org.invalid> posted:

A closely related puzzle is to arrange FIVE distinct points such that
only two inter-point distances occur.

... and describe the weird/wonderful relationship between this 5-point companion puzzle and the question I asked earlier:

Which of the six {4-point, 2-distances} arrangements is hardest to think of?

I think this 5-point puzzle is likely to be easier for many people than finding the 6'th solution to
the 4-point puzzle, and might lead them to the 6'th solution indirectly! (That's amusing when you
think about it...)

Anyway, nice puzzle!

Mike.

Yes. The 5-point solution is easier to think of than the 4-point solution.

Are there other examples where an ostensibly harder puzzle is in practice easier to solve than a less general, ostensibly easier-to-solve subset
of the puzzle?

Cheers,
noU b-Anoab!+ noe

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From Newsgroup: rec.puzzles


James Dow Allen <user4353@newsgrouper.org.invalid> posted:

Mike Terry <news.dead.person.stones@darjeeling.plus.com> posted:

On 20/03/2026 17:11, James Dow Allen wrote:

Now: Solve the companion puzzle:
James Dow Allen <user4353@newsgrouper.org.invalid> posted:

A closely related puzzle is to arrange FIVE distinct points such that
only two inter-point distances occur.

... and describe the weird/wonderful relationship between this 5-point companion puzzle and the question I asked earlier:

Which of the six {4-point, 2-distances} arrangements is hardest to think of?

I think this 5-point puzzle is likely to be easier for many people than finding the 6'th solution to
the 4-point puzzle, and might lead them to the 6'th solution indirectly! (That's amusing when you
think about it...)

Anyway, nice puzzle!

Mike.

Yes. The 5-point solution is easier to think of than the 4-point solution.

Are there other examples where an ostensibly harder puzzle is in practice easier to solve than a less general, ostensibly easier-to-solve subset
of the puzzle?

Cheers,
noU b-Anoab!+ noe

What is that supposed to be in the middle? it sort-of reads like GUI


4-point and 5-point problems -- are these old? Did Euclid know them?
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From Newsgroup: rec.puzzles

On 23/03/2026 16:16, HenHanna@NewsGrouper wrote:

James Dow Allen <user4353@newsgrouper.org.invalid> posted:

Mike Terry <news.dead.person.stones@darjeeling.plus.com> posted:

On 20/03/2026 17:11, James Dow Allen wrote:

Now: Solve the companion puzzle:
James Dow Allen <user4353@newsgrouper.org.invalid> posted:

A closely related puzzle is to arrange FIVE distinct points such that >>>>> only two inter-point distances occur.

... and describe the weird/wonderful relationship between this 5-point >>>> companion puzzle and the question I asked earlier:

Which of the six {4-point, 2-distances} arrangements is hardest to think of?

I think this 5-point puzzle is likely to be easier for many people than finding the 6'th solution to
the 4-point puzzle, and might lead them to the 6'th solution indirectly! (That's amusing when you
think about it...)

Anyway, nice puzzle!

Mike.

Yes. The 5-point solution is easier to think of than the 4-point solution. >>
Are there other examples where an ostensibly harder puzzle is in practice
easier to solve than a less general, ostensibly easier-to-solve subset
of the puzzle?

Cheers,
noU b-Anoab!+ noe

What is that supposed to be in the middle? it sort-of reads like GUI

4-point and 5-point problems -- are these old? Did Euclid know them?

I don't know if Euclid knew of the puzzle.

Of the 6 solutions to the 4-point puzzle, 5 of them have very obvious geometric constructions using
straightedge and compass. E.g. we all know how to construct a square, right? And one of the
solutions is the three vertices of an equilateral triangle together with its centre point - and we
all know how to construct an equilateral triangle, and find its centre by bisecting its angles or
edges... And so on, but /one/ of the 6 4-point solutions does not have such an easy geometric
construction!

That's not to say it doesn't have such a construction - it does, and Euclid knew how to perform that
construction, although he might not have known of its relation to the 4-point puzzle (or even have
heard of that puzzle). That's the only reason I mentioned Euclid.

[To fully appreciate what I'm saying, I'm afraid you'll have to find the 6'th solution for yourself!
Then you can try to construct it with straightedge+compass and submit your solution for extra
marks if you can do it! :)]

Here's my attempt at forging James' signature:

noU b-Anoab!+ noe

hmm, looks pretty close... (I don't know what it means though)

Mike.

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From Newsgroup: rec.puzzles


Mike Terry <news.dead.person.stones@darjeeling.plus.com> posted:

I don't know if Euclid knew of the puzzle.

I doubt it. I don't remember how I learned of it some years ago.
Most probably I either invented it myself, or found it right here
in rec.puzzles!

Of the 6 solutions to the 4-point puzzle, 5 of them have very obvious geometric constructions using
straightedge and compass....
but /one/ of the 6 4-point solutions does not have such an easy geometric construction!

[To fully appreciate what I'm saying, I'm afraid you'll have to find the 6'th solution for yourself!
Then you can try to construct it with straightedge+compass and submit your solution for extra
marks if you can do it! :)]

I think I constructed it 60 years ago, but I'm too lazy (and fearful of failure!) to even attempt it with my fast-fading cerebrum. Two comments:

(1) It was Carl Friedrich Gauss who proved that a regular n-gon could be constructed if n's factors are all Fermat primes or 2. This is called the GaussrCoWantzel theorem since Pierre Wantzel proved the 'only if' part a few decades later.

(2) A "construction" method I invented is to start with a narrow paper strip
of constant width and carefully tie a simple knot, tightening the knot
without bending the strip. Presto! The desired shape.

Here's my attempt at forging James' signature:

noU b-Anoab!+ noe

hmm, looks pretty close... (I don't know what it means though)

Mike.

I've lost interest in changing my name since learning that my e-mail
server will not allow this as an account name. The rot13 name
signed below is boring. I read about someone who changed his name
to ZD57439 because a change-of-name was less expensive than personalized license plate. I'll keep looking....

Purref,
Wnzrf
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From Newsgroup: rec.puzzles

On 3/25/2026 4:44 AM, James Dow Allen wrote:

I've lost interest in changing my name since learning that my e-mail
server will not allow this as an account name. The rot13 name
signed below is boring. I read about someone who changed his name
to ZD57439 because a change-of-name was less expensive than personalized license plate. I'll keep looking....

Purref,
Wnzrf

You might consider signing with an anagram (e.g., "OLD MAN'S A JEWEL").
--
- A CLOWN GRIN <:oD


--
This email has been checked for viruses by AVG antivirus software.
www.avg.com
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From Newsgroup: rec.puzzles


"Carl G." <carlgnews@microprizes.com> posted:

On 3/25/2026 4:44 AM, James Dow Allen wrote:

I've lost interest in changing my name since learning that my e-mail
server will not allow this as an account name....

You might consider signing with an anagram (e.g., "OLD MAN'S A JEWEL").

I'd rather just change my name than be reminded I'm OLD.

This is NOT the first time a Usenetter has anagrammed my name to 'OLD.'
When I was in my 60's I could almost credibly shrug this off: I was "middle-aged", but I think the window for that euphemism has expired
by now. (Didn't I see some of you guys here in a previous century?
Some of you might be middle-aged by now yourselves.)

It's the 'DOW' that causes the trouble of course (tho I don't mind 'JEWEL')
and I don't know where it even came from. My father was a DOW,
as was his father, and as was that grandfather's uncle. But if the often-mispronounced syllable had any special significance it's long
forgotten.

Instead of 13 letters, my birth certificate uses 16 letters!
Even harder to pronounce than 'DOW' is the 'III' appended to my name there. (Could this be why I'm one of the fogeys that still refers to the
Three of Spades as the 'Trey'?)

If you can't think of a more flattering anagram using the added three I's, perhaps I'll just find a moniker I like and anagram backwards to form
my new name.

Or, spelling 'III' with a lower-case L ('IIl'),
I could valedict with this gibberish:

Cheers
wILl smIle on a JADe
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From Newsgroup: rec.puzzles

On 3/26/2026 12:58 AM, James Dow Allen wrote:

"Carl G." <carlgnews@microprizes.com> posted:

On 3/25/2026 4:44 AM, James Dow Allen wrote:

I've lost interest in changing my name since learning that my e-mail
server will not allow this as an account name....

You might consider signing with an anagram (e.g., "OLD MAN'S A JEWEL").

I'd rather just change my name than be reminded I'm OLD.

This is NOT the first time a Usenetter has anagrammed my name to 'OLD.'
When I was in my 60's I could almost credibly shrug this off: I was "middle-aged", but I think the window for that euphemism has expired
by now. (Didn't I see some of you guys here in a previous century?
Some of you might be middle-aged by now yourselves.)

It's the 'DOW' that causes the trouble of course (tho I don't mind 'JEWEL') and I don't know where it even came from. My father was a DOW,
as was his father, and as was that grandfather's uncle. But if the often-mispronounced syllable had any special significance it's long forgotten.

Instead of 13 letters, my birth certificate uses 16 letters!
Even harder to pronounce than 'DOW' is the 'III' appended to my name there. (Could this be why I'm one of the fogeys that still refers to the
Three of Spades as the 'Trey'?)

If you can't think of a more flattering anagram using the added three I's, perhaps I'll just find a moniker I like and anagram backwards to form
my new name.

Or, spelling 'III' with a lower-case L ('IIl'),
I could valedict with this gibberish:

Cheers
wILl smIle on a JADe

I was also involved in rec.puzzles in the last millennia.
Sometimes "old man" refers to one's father. If your "old man" passed on
his treasured "DOW" heritage to you, then maybe your old man's a jewel.
--
Carl Ginnow (A CLOWN GRIN)


--
This email has been checked for viruses by AVG antivirus software.
www.avg.com
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From Newsgroup: rec.puzzles


"Carl G." <carlgnews@microprizes.com> posted:

I was also involved in rec.puzzles in the last millennia.

(Nitpick: millenia is a PLURAL. In my dialect you would need to be
at least 1027 years old to have been involved in previous millennia.)

I was active at Usenet during the late 1980's but not so much in the 1990's.

I Do recall that you were probably the very BEST puzzle composer here at r.p during the early years of this millennium. In particular one of your puzzles intrigued me enough that I composed a webpage describing it, and giving the source code I eventually wrote to solve this puzzle
(which I call "Ginnow's Solitaire" or "Ginnow's Sieve").:
https://fabpedigree.com/james/gsieve.htm

Searching just now I didn't find your post(s) on this puzzle though
I see I mentioned it in another thread in 2010.


Pleasant valedictions from the poster formerly (and very briefly) known as noUb-Anoab!+noe
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From Newsgroup: rec.puzzles


James Dow Allen <user4353@newsgrouper.org.invalid> posted:

"Carl G." <carlgnews@microprizes.com> posted:

I was also involved in rec.puzzles in the last millennia.

I Do recall that you were probably the very BEST puzzle composer here at r.p during the early years of this millennium. In particular one of your puzzles intrigued me enough that I composed a webpage describing it, and giving the source code I eventually wrote to solve this puzzle
(which I call "Ginnow's Solitaire" or "Ginnow's Sieve").:
https://fabpedigree.com/james/gsieve.htm

I located the thread. It was "The 101 Game", started by Carl G. on 31 May 2000.

Does this date lie in the present millennium or an earlier one?
Has the political schism ever been healed which divided two camps:
(A) Traditionalists who believe that one (1) is the first (1st) number.
(B) The "Turning Odometer" fans, perhaps joined by a coalition of C programmers.


JDA III
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